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- W2983643934 abstract "In 1909, Hardy gave an example of a transcendental entire function, $f$, with the property that the set of points where $f$ achieves its maximum modulus, $mathcal{M}(f)$, has infinitely many discontinuities. This is one of only two known examples of such a function. In this paper we significantly generalise these examples. In particular, we show that, given an increasing sequence of positive real numbers, tending to infinity, there is a transcendental entire function, $f$, such that $mathcal{M}(f)$ has discontinuities with moduli at all these values. We also show that the transcendental entire function lies in the much studied Eremenko-Lyubich class. Finally, we show that, with an additional hypothesis on the sequence, we can ensure that $f$ has finite order." @default.
- W2983643934 created "2019-11-22" @default.
- W2983643934 creator A5014784744 @default.
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- W2983643934 date "2020-06-29" @default.
- W2983643934 modified "2023-10-18" @default.
- W2983643934 title "Variations on a theme of Hardy concerning the maximum modulus" @default.
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- W2983643934 doi "https://doi.org/10.1112/blms.12387" @default.
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